Poly-Time Knot Theory and Quantum Algebra

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One of the major triumphs of mathematics in the 1980s, which lead to at least 3 Fields medals (Jones, Drinfel'd, Witten) was the unexpected realization that low dimensional topology, and in particular knot theory, is closely related to quantum field theory and to the theory of quantum groups. Knot theory is mundane and ages-old; anything "quantum" seems hyper-modern. Why would the two have anything to do with each other?The answer is long and complicated and has a lot to do with the "Yang-Baxter Equation" (YBE). The YBE on the one hand can be interpreted in knot theory as "the third Reidemeister move", or as "controlling the most basic interaction of 3 pieces of string" (this turns out to be a very crucial part of knot theory). On the other hand solutions of the YBE arise from "quantum" machinery. Hence the quantum is useful to the knotted, and by similar ways, to the rest of low dimensional topology.But "quantum" has a caveat, which makes it super-exciting (to some) yet bounds its usefulness (to others). When quantum systems grow large (as they do when the knot or low-dimensional space we study grows complicated), their "state space" grows at an exponential rate. "Quantum computers" aim to exploit this fact and make large quantum systems performs overwhelmingly large computations by utilizing their vast state spaces. But quantum computers aren't here yet, may take many years to come, suffer from other limits on what they can do, and much of low-dimensional topology is anyway outside of these limits. So at least for now and likely forever, many things that have "quantum" in their description are exponentially-complex to compute, which in practice means that they cannot be computed beyond a few simple cases.Recently Van der Veen and myself, following Rozansky and Overbay, found a corner (figuratively speaking) of the vast state space of the quantum machinery used in knot theory, which can be described in just polynomial complexity, and which carries enough information to still speak to knot theory. The "knot invariants" constructed that way seem to be the strongest invariants we know that are computable even for very large knots.Our approach utilizes the fact that complicated symmetry groups often have much simpler "contractions". A well known example is the Lorentz group of relativity theory, which at small velocities contracts to the Galilean group of classical mechanics. In a similar manner we find that the symmetry algebras underlying the useful solutions of the Yang-Baxter equation, namely semi-simple algebras such as sl(n), have contractions that are "solvable algebras", and that the same operations that are exponentially complex for the original sl(n) symmetry become polynomially-complex (namely, much simpler) within and near these solvable contractions.Much remains to be done: implementation, documentation, application, generalization. I hope to achieve all that over this 5-year grant period.